Combine pineapple weed and water in a small pot, cover and bring to a boil over high heat. Reduce heat to medium-low and simmer for 10 minutes.
Remove pot from heat and allow mixture to steep for about 20 minutes.
Strain into a clean pot through a sieve lined with cheesecloth. Twist the cheesecloth into a bundle and squeeze to extract as much liquid as possible.
Add 1 cup of sugar to the pot. Bring to the boil and cook just until the sugar is dissolved. Remove from heat. For a stronger flavour, tie off the bundle of pineapple weed and soak it in the syrup while it cools. Squeeze the bundle to extract the liquid before you pour into jars.
Once the syrup is at room temperature, pour it into a clean jar and refrigerate. Will keep for up to one month.